Polarization matrices in anisotropic heterogeneous

elasticity

S.A. Nazarov1, J. Sokolowski2 and M. Specovius-Neugebauer3 ∗

1 Institute of Mechanical Engineering Problems, Russian Academy of Sciences, Saint-Petersburg,

Russia

email: srgnazarov@yahoo.co.uk

2 Institut Elie Cartan, Laboratoire de Mathematiques, Universite Henri Poincare

Nancy 1, B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France

email: Jan.Sokolowski@iecn.u-nancy.fr, url: http://www.iecn.u-nancy.fr/ sokolows/

3: Universitat Kassel, Heinrich Plett Str. 40, D-34132 Kassel, Germany

email: specovi@mathematik.uni-kassel.de, url: http://www.mathematik.uni-kassel.de/ specovi/

Abstract

Polarization matrices (or tensors) are generalizations of mathematical objects likethe harmonic capacity or the virtual mass tensor. They participate in many asymptoticformulae with broad applications to problems of structural mechanics. In the presentpaper polarization matrices for anisotropic heterogeneous elastic inclusions are inves-tigated, the ambient anisotropic elastic space is allowed to be inhomogeneous near theinclusion as well. By variational arguments the existence of unique solutions to the cor-responding transmission problems is proved. Using results about elliptic problems indomains with a compact complement, polarization matrices can be properly defined interms of certain coefficients in the asymptotic expansion at infinity of the solution to thehomogeneous transmission problem. Representation formulae are derived from whichproperties like positivity or negativity can be read of directly. Further the behavior ofthe polarization matrix is investigated under small changes of the interface.

Keywords: Asymptotic analysis, polarization matrix, elastic moment tensors, anisotropicelasticity problems, transmission conditionsAMS classification: 35Q30, 49J20, 76N10

1 Introduction

The asymptotic analysis in singularly perturbed geometrical domains is a tool of math-ematical modeling in elasticity or for coupled models e.g., in the piezoelectricity and inthe fluid-structure interaction. Asymptotic analysis makes it possible to derive an approx-imation of solutions to the complex (complicated) PDE’s models by means of solutions tosimpler PDE’s models in geometrical domains which are more attractive e.g., from numericalpoint of view. It permits to include singular variations of the underlaying domains whichis very convenient for the analysis as well as for the numerics of optimization and inverseproblems. This is documented, e.g. in optimal design problems for structure mechanics bywide applications of the so-called topological derivatives of shape functionals. In this con-text, formulae which can be really used in modeling and computation seem to be important.

∗This paper was prepared during visits of S.A. Nazarov to the university of Kassel and to the InstituteElie Cartan of the University Henri Poincare Nancy 1. The research of S.A.N. is partially supported by thegrant RFFI-09-01-00759; J.S. is partially supported by the Projet CPER Lorraine MISN: Analyse, optimi-sation et controle 3 in France, and the grant N51402132/3135 Ministerstwo Nauki i Szkolnictwa Wyzszego:Optymalizacja z wykorzystaniem pochodnej topologicznej dla przeplywow w osrodkach scisliwych in Poland,M.S. is partially supported by the DFG in the frame of the collaborative research center SFB/TR ”Prozess-integrierte Herstellung funktional gradierter Strukturen auf der Grundlage thermo-mechanisch gekoppelterPhanomene”

1

In this paper we assemble results for the polarization matrices in anisotropic elasticity [46].They are generalizations of classical objects in harmonic analysis, namely quadratic formsassociated to polarization tensors and virtual masses (see [55]).

The motivation to deal with these objects is best explained by their multifaceted appli-cations such as

• the detection of imperfections in an homogeneous medium;

• problems of damage and fracture;

• mechanics of composites, especially dilute composites, i.e., with sparsely distributeddefects and inclusions;

• vibration of inhomogeneous bodies;

• shape optimization

Any asymptotic (approximative) formula for the stress-strain state of an elastic body withsmall, sparsely placed defects includes the so-called defect polarization matrices. Apparentlythese integral attributes were introduced and analyzed more than 20 years ago in the con-text of three and two dimensional isotropic elasticity theory in [36, 37], where polarizationmatrices were used in the asymptotic analysis of the stress-strain state of a defected solid.Polarization matrices are employed to describe asymptotic properties of elastic bodies withsmall inclusions, holes or cavities, voids and cracks, also in anisotropic elastic materials [46].There are generalizations of this concept, see e.g. [45, 16, 54] and [[52]; Chapter 5] whereanalogous matrices are defined for general elliptic systems.

Polarization tensors as useful tools in analysis and description of effective properties ofdilute composites, which leads to approximate but explicit formulae, were treated in thebooks [25, 17, 30, 32, 2, 4], and papers [44, 9, 31]. Mathematical definitions of damagetensors and measures in relation with polarization tensors were given in [46, 49, 50].

Concerning fracture mechanics we refer to the classical paper of Griffith [15] which is infact devoted to the evaluation of the polarization matrix for a crack located in the isotropicplane. Actually the matrix enters asymptotic formulae for the variation of energy of defor-mation which govern the nucleation or the growth for a straight crack (cf. [26]). There aremany papers on similar problems in three spatial dimensions, and also on a description ofinteractions between a main crack with some defects of different nature (see e.g., the papers[36, 33, 34, 35, 47] and the books [22, 25, 30]). We point out, that curving and kinkingof cracks is described by matrices which are fully analogous to the concept of polarizationmatrices cf. [53, 9, 43].

We mention that in spectral problems with small singular perturbations of the boundarythe leading term in the asymptotics of the eigenvalues can be calculated as a function of thepolarization matrix or similar objects (see the papers [27, 12, 13, 42, 6] and the books [25],[30]1)

In shape optimization the polarization matrices are one of the main ingredients of thetopological derivatives of shape functionals [54], even if it is not explicitly acknowledged.In particular, in shape optimization and identification problems for eigenvalues, the role ofpolarization matrices seems to be premondial [42].

During the last decade polarization matrices has undergone a revival in the frameworkof certain questions related to numerics. In [5, 3] polarization tensors related to two-dimensional scalar elliptic equations were used to characterize the effective conductivityof dilute composite materials which are anisotropic with respect to their electrical proper-ties. In [7, 3, 4] the theory of layer-potentials was used to investigate integral characteristicsfor penetrable inclusions isotropic elasticity problems, there the authors called them elasticmoment tensors. Polarization matrices for scalar problems and higher order EMTs 2 inisotropic materials are analyzed and various applications including inverse problems such asthe detection of inclusions are presented in the books [3, 4], but has also been investigatedalready in [17] and other works.

1In paper [38] there is an error which is repeated in [30] and it is corrected in [13].2See further comments on higher order polarization matrices in Remark 2.10.

2

Evidently, the polarization matrices (tensors) are quite important for modeling in solidmechanics. Unfortunately, the matrices are known explicitly only for some specific canonicalshapes, e.g. among others for a ball and an ellipse, for an elliptic plane crack in three spatialdimensions and a straight crack in two spatial dimensions. The famous theorem of Eshelbyon elliptic inclusions ([14, 19] and others) provides simple but implicit algebraic formulae,with broad applications in the literature in mechanics. Let us point out that the resultspresented in [28, 29] and reproduced in [30] on the explicit representations of polarizationmatrices for arbitrary shaped cavities in the isotropic plane, are wrong. A mistake is foundand is discussed in [8], however, after correction the problem reduces to solving an infinitesystem of algebraic equations and then, sadly enough, makes the correct formulae completelyimplicit.

In the present paper, we analyze polarization matrices for two and three dimensionaltransmission problems problems in anisotropic elasticity. We further use methods of asymp-totic analysis and shape sensitivity analysis to investigate the behavior of the polarizationmatrix under changes of the interface.

Since the fundamental solution matrix is not known explicitly for general three dimen-sional anisotropies, moreover we allow for nonconstant coefficients in the elastic inclusionas well as for the elastic space, the techniques of integral equations as used e.g. in [3] (andthe other quoted papers by these authors) are not available. However, the existence resultsneeded in this context can be obtained by comparably simple variational arguments evenfor Lipschitz continuous interfaces. The integral attributes leading to the polarization ma-trices can be defined by using results for asymptotic representations for elliptic problems indomains with conical outlets to infinity.

In the present paper we derive representation formulae from which properties like sym-metry or positivity can be read of directly by comparison of the elastic properties of theinclusion and the ambient elastic space.

The plan of this paper is as follows: We start some basic notations such as function spaces,the general assumptions and the formulation of the problem in Section 1.1. We use thematrix/column notation - the so called Voigt notation - for the constitutive laws in elasticity(see [48, 11]). In Section 2 we present the solvability of elastic transmission problems inthree spatial dimensions where polarization matrices appear as coefficients in the asymptoticrepresentation at infinity of certain solutions. Most of the attention is paid to describe thematrices in the form of intrinsic integral and energy attributes of the elastic inclusions. Suchrepresentations provide a description of general properties, e.g., positiveness/negativeness(see Theorems 2.7 and 2.8) and are very useful in the shape sensitivity analysis of Section3.

Note that the concept of algebraically equivalent media (cf. [10, 18] for two dimensionalcase and [20, 21] in three dimensions) allows for simple formulae of the entries of polarizationmatrices, but only in the case of very specific shapes and stiffness tensors (see Section 2.5).

Section 3 is devoted to the behavior of the polarization matrices under small changes ofthe interfaces. The benefits of these results are twofold. First, if the polarization matrixis known for a specific shape, we can use the shape derivatives in order to evaluate thematrix for a regular perturbation of the shape. The second application concerns the shapeoptimization, identification and design in structural mechanics. We can minimize/maximizea specific functional with respect to the polarization matrices, some applications in inverseproblems can be found e.g. in [3], but it is still the field of current research. There areat least two possible approaches to the shape sensitivity analysis. One is based on theasymptotic analysis in singularly perturbed domains, the method is explained in details forthe elasticity in two dimensions. The second method uses the boundary variation technique[57] for the shape functionals which resemble the energy functionals. In this way we derivedirectly the shape derivatives of the polarization matrices written in the form of a minimizerof the given functional, or, equivalently, evaluate the material derivatives of the minimizersand obtain the same expression for the shape derivatives of the polarization matrices.

3

1.1 Basic notations, elasticity problems in Voigt notation

We start with introducing some geometrical notations and standard auxiliary results. Theball and the sphere of radius R > 0 are denoted by BR and SR, respectively.

Let Ω ⊂ R3 be a Lipschitz domain with a compact connected complement Ω• andboundary Γ. For x ∈ Γ, we denote by n(x) the outward (with respect to Ω) unit normalvector. For integer l, H l(Ω), H l(Ω•) and – in case of smooth Γ – the spaces Hs(Γ) fors ∈ R are the usual Sobolev-Slobodetskii spaces (see [23], e.g.). If Γ is only Lipschitz, thenat least the trace space H1/2(Γ) is well defined together with a continuous trace operatorγ : H1(Ω•) → H1/2(Γ), such that γφ = φ|Γ for smooth functions. We keep the notationφ|Γ also for H1-functions. The trace operator possesses a continuous right inverse: Forφ ∈ H1/2(Γ) there exists Φ• ∈ H1(Ω•) such that Φ•|γ = φ and

‖Φ•; H1(Ω•)‖ 6 C(Γ)‖φ; H1/2(Γ)‖.Further we use the notation ( , )Ξ for the scalar product in L2(Ξ) for various suitable setsΞ ⊂ R3.

Formulation of the transmission problem. We recall the so-called Voigt notation, i.e.matrix-vector notation for the elasticity problems [52, 48, ?]. This means we think of thedisplacement vector u = (u1, u2, u3)> as a column in R3 and introduce the strain column ofheight 6

ε(u) = (ε11(u), ε22(u), ε33(u),√

2ε23(u),√

2ε31(u),√

2ε12(u))>

where α = 2−1/2 and εjk(u) are Cartesian components of the strain tensor given by

εjk(u) =12

(∂uj

∂xk+

∂uk

∂xj

), j = 1, 2,

with j, k = 1, 2, 3. Then we have ε(u) = D(∇)u, where

D(ξ)> =

ξ1 0 0 0 αξ3 αξ2

0 ξ2 0 αξ3 0 αξ1

0 0 ξ3 αξ2 αξ1 0

, α =

1√2, ξ ∈ R3 . (1.1)

Analogously we define the stress columns σ(u) and σ•(u) in Ω and Ω•, respectively, thenHooke’s law can be expressed as

σ(u) = Aε(u), σ•(u) = A•ε(u). (1.2)

where A and A• are symmetric and positive definite 6 × 6 matrices, the stiffness matrices,containing the elastic moduli of the material in Ω and Ω•, respectively.

Remark 1.1 For the reader’s convenience, we recall the relation between the usual Hooketensor and the stiffness matrix (see e.g.,[48, Sec. 2.2] or [11, p.42].) In tensor notation,Hooke’s law reads

σij =∑p,q

aijpqεpq, i, j = 1, 2, 3. (1.3)

We recall that symmetry for a rank four tensor means

aijpq = aji

pq = apqij .

The matrix A in (1.2) gets the form

A =

a1111 a11

22 a1133

√2a11

23

√2a11

31

√2a11

12

a2211 a22

22 a2233

√2a22

23

√2a22

31

√2a22

12

a3311 a33

22 a3333

√2a33

23

√2a33

31

√2a33

12√2a23

11

√2a23

22

√2a23

33 2a2323 2a23

31 2a2312√

2a3111

√2a31

22

√2a31

33 2a3123 2a31

31 2a3112√

2a1211

√2a12

22

√2a12

33 2a1223 2a12

31 2a1212

(1.4)

In a similar manner, the tensor for two space dimensions is transformed into a 3× 3-matrix.

4

We assumeA(x) = A0 + Ae(x) with Ae

ij ∈ C10 (Ω), (1.5)

i.e. in particular Ae(x) = 0 for r = |x| large, say r ≥ R0, where R0 large enough such thatΩ• is contained in the open ball BR0 . The matrix A0 is constant and positive definite, i.e.the elastic space is homogeneous far away from the inclusion. However, the inclusion againcan be heterogeneous, we suppose that the entries of A• are C1- functions in Ω•, e.g.. Boththe matrices A(x) and A•(x) must be uniformly positive definite, i.e.

ξ>A(x)ξ ≥ a|ξ|2 , ξ>A•(x)ξ ≥ a•|ξ|2

holds for any vector ξ ∈ R6 and all x ∈ Ω and x ∈ Ω• with positive constants a, a•. Theelasticity problem in Ω ∪ Ω• reads as follows:

D(−∇)>A(x)D(∇)u(x) = f(x), x ∈ Ω = R3 \ Ω•;

D(−∇)>A•(x)D(∇)u•(x) = f•(x), x ∈ Ω•(1.6)

To shorten the notations, we denote the differential operators (1.6) by L and L• respectively.In order to formulate the transmission conditions we need the associated Neumann operators

Nu : = D(n(x))>A(x)D(∇)u(x),

N •u• : = D(n(x))>A•(x)D(∇)u•.(1.7)

Using these notations, problem (1.6) is supplied with the boundary conditions on Γ:

u− u• = g0, Nu(x)−N •u•(x) = g1. (1.8)

If f decays sufficiently fast we may add the condition at infinity

|u(x)| = O(|x|−1), as |x| → ∞. (1.9)

Here f , f• are volume forces while g0 and g1 stand for jumps of displacements and tractionson the interface Γ.

Note that the two-dimensional problem can be formulated in just the same way, withstress and strain columns (σ11, σ22,

√2σ12)>, (ε11, ε22,

√2ε12)>in R3 instead of R6 and 3×3-

Hooke matrices, while the matrix D(ξ) in (1.1) has to be replaced by

D(ξ) =

ξ1 00 ξ2

2−1/2ξ2 2−1/2ξ1

2 The polarization matrices for three-dimensional anisotropic elas-ticity problems

2.1 Solvability of the transmission problems.

We look for a variational formulation of this problem. By H, we denote the completion ofC∞0 (R3)3 with respect to the “energy” norm

‖D(∇)u; L2(R3)‖. (2.1)

Using the Fourier transform and Hardy’s inequality, we obtain the Korn’s inequality

‖(1 + r)−1u; L2(R3)‖+ ‖∇u; L2(R3)‖ 6 1√10‖D(∇)u; L2(R3)‖ u ∈ H. (2.2)

In this context we recall the definition of Kondratiev norms, adapted to the special caseswe need here: Let G ⊆ R3 an (unbounded) domain, β ∈ R and l ∈ N0 = 0, 1, . . . be fixed.Then the Kondratiev space V l

β(G) consists of all u ∈ H lloc(G) such that the norm

‖u; V lβ(G)‖ =

( l∑

k=0

‖(1 + |x|)β−l+k∇kx u; L2(G)‖2

)1/2

< ∞.

5

Note that the weight control the behavior at infinity of the functions under consideration.Estimate (2.2) implies that H = V 1

0 (R3) and the norms are equivalent.For sufficiently smooth vector fields u and v we have Green’s formulae on Ω and Ω•:

(Lu, v)Ω + (Nu, v)Γ = (AD(∇)u,D(∇)v)Ω,

(L•u, v)Ω − (N •u, v)Γ = (A•D(∇)u, D(∇)v)Ω• , (2.3)

observe that n is the internal normal vector on Γ with respect to Ω•. That is why thereappears the sign minus in (2.3). In particular, for any vector function u ∈ H2

loc(Ω), u• ∈H2(Ω•), satisfying (1.6) and (1.9), and v ∈ C∞0 (R3), the addition of the two formulae leadsto

(AD(∇)u,D(∇)v)Ω + (A•D(∇)u•, D(∇)v)Ω• = (f•, v)Ω• + (f, v)Ω + (g1, v)Γ (2.4)

If g0 = 0 and u ∈ V 21 (Ω) in the situation above, then we can glue u, u• together and obtain

a vector field w ∈ V 10 (R3) = H. Furthermore, due to our assumptions on the matrices A, A•

and Korn’s inequality (2.2), the left hand side of this equality defines a scalar-product b( , )on H which induces a norm equivalent to the energy norm defined by (2.1). If f and f• arerestrictions of F ∈ V 0

1 (Ω) the right hand side of (2.4) defines a continuous linear functionalon H, even in case (g1, v)Γ being replaced by 〈g1, v〉 with g1 ∈ H−1/2(Γ). Thus, for g0 = 0,equation (2.4) takes the form

b(u, v) = F (v) with F ∈ H′. (2.5)

Clearly, Neumann boundary values in general do not exist for u ∈ H. Since L,L• aredifferential operators in divergence form, we may use a well known weak trace theorem toovercome this difficulty (see [58, p. 9]).

Proposition 2.1 For u ∈ H1(Ω•)3 with L•u ∈ L2(Ω)3, there exists a trace N •u ∈ H−1/2(Γ),such the Green’s formula (2.3) is valid and the following estimate holds independent of u:

‖N •u; H−1/2(Γ)‖ 6 C(‖D(∇)u; L2(Ω)‖+ ‖L•u;L2(Ω)‖) . (2.6)

An analogous result is true for u ∈ V 10 (Ω) with Lu ∈ V 0

1 (Ω).

To rewrite the general problem (1.6) – (1.8) in the form (2.5), it is necessary to reduce itto the case g0 = 0. The obvious way is here to search for u• = U•−G, where G is a suitableextension of g0 onto Ω•. If g0 ∈ H3/2(Γ), this works with any extension G ∈ H2(Ω•). Forg0 ∈ H1/2 we take the unique weak solution G ∈ H1(Ω) to the problem

L•G = 0 in Ω•, G = g0 on Γ. (2.7)

Here we have the estimate

‖G; H1(Ω•)‖ 6 C‖g0;H1/2(Γ)‖ (2.8)

with a constant independent of u0, and by regularity results for elliptic problems [59] thisextends to

‖G; H l+2(Ω•)‖ 6 C‖g0; H l+3/2(Γ)‖, l ∈ N0, (2.9)

provided the surface Γ as well as the coefficient functions in A• are sufficiently smooth andg0 ∈ H l+3/2(Γ).

Definition 2.2 Let f• ∈ L2(Ω•)3, f ∈ V 01 (Ω)3, g0 ∈ H1/2(Γ) and g1 ∈ H−1/2(Γ)3 be given,

and let G ∈ H1(Ω•)3 be the extension of g0 into Ω• defined by (2.7). We call a pair u, u•of vector fields defined on Ω and Ω•, respectively, a weak solution of the boundary valueproblem (1.6) – (1.8), if u, u• + G ∈ H and following identity is fulfilled:

(AD(∇)u,D(∇)v)Ω + (A•D(∇)u•, D(∇)v)Ω• = (f•, v)Ω• + (f, v)Ω + 〈g1, v〉Γ (2.10)

6

Standard Hilbert space arguments (the Riesz representation theorem) lead to the follow-ing result.

Proposition 2.3 Problem (2.4) has a unique weak solution, and the following estimateholds:

‖u;V 10 (Ω)‖+ ‖u•; H1(Ω•)‖

6 c(‖f•;L2(Ω•)‖+ ‖f ; V 0

1 (Ω)‖+ ‖g0;H1/2(Γ)‖+ ‖g1; H−1/2(Γ)‖)

. (2.11)

If the surface Γ and the matrix functions A, A• are smooth and for some l ∈ N = 1, 2, . . . we have

f ∈ V l−1l (Ω)3, f• ∈ H l−1(Ω•)3,

g0 ∈ H l+1/2(Γ)3, g1 ∈ H l−1/2(Γ)3,(2.12)

then u ∈ V l+1l (Ω)3, u• ∈ H l+1(Ω•)3 and the pair u, u• is a strong solution to the elliptic

transmission problem (1.6), (1.9). Moreover, the estimate

‖u;V l+1l (Ω)‖+ ‖u•; H l+1(Ω•)‖ 6 cFl (2.13)

holds true where Fl denotes the sum of the norms of the data (2.12) in the spaces indicated.

Proof. Due to Proposition 2.6 the trace N •G ∈ H−1/2(Γ), and

(A•D(∇)G,D(∇)v)Ω• + 〈N •G, v〉Γ = 0.

Thus (2.10) is fulfilled if U = u, u• + G solves

b(U, v) = (f•, v)Ω• + (f, v)Ω + 〈g1 −N •G, v〉Γ (2.14)

for all v ∈ H. Clearly, (2.14) is again of the form (2.5), thus application of the Rieszrepresentation theorem for a linear functional in a Hilbert space ensures the existence of aunique solution while estimates (2.2) and (2.8) lead to estimate (2.11). The estimate (2.13)follows then from (2.9) and regularity results for elliptic problems (see, e.g., [56]). 2

Remark 2.4 Of course, equation (2.5) is uniquely solvable for any F ∈ H′, and we obtain

‖u;H‖ 6 ‖F ;H′‖. (2.15)

However, for general u ∈ H, the second transmission condition is senseless, and it is obviousthat there exist functionals F ∈ H which are not of the form (2.14) with f•, f and g0, g1

as in Definition 2.2. At the same time is known that any continuous linear functional onV 1

0 (R3) has a (non unique) representation as

F (v) = (f, v)R3 +3∑

i=1

(Fi, ∂iv)R3 , (2.16)

with f ∈ V 01 (R3), Fi ∈ L2(R3), and

‖F ; V 10 (R3)′‖2 = inf

‖f ;V 01 (R3)‖2 + ‖Fi; L2(R3)‖2 , (2.17)

where the infimum is taken over all representations. The solution u to (2.5) is independentof f and Fi appearing in (2.16), of course, as long as they lead to the same functional. 2

We can introduce a weak transmission problem: For given F ∈ H′, g0 ∈ H1/2(Γ) let Gbe the solution to (2.7). We call u, u• a solution to the weak transmission problem (1.6)and (1.9)2, if U = u, u• + G ∈ H and U solves (2.5). Such a solution always exists andfrom (2.8), (2.15) and (2.17) we obtain

‖u;V 10 (Ω)‖+ ‖u•;H1(Ω•)‖6 c

( ‖f ; V 01 (R3)‖+

∑

i

‖Fi; L2(R3)‖+ ‖g0; H1/2(Γ)‖ ).

Again it is not possible to subtract an arbitrary prolongation G ∈ H1(Ω•) from u•, sincethen Green’s formula (2.3) leads to a boundary distribution which is contained in H−3/2(Γ)only. Insofar the weak transmission problem is equivalent to find in parallel weak solutionsto the problems (2.7) and L u = F in R3, where L is the differential operator ”glued”together from L• and L. 2

7

2.2 Asymptotic behavior of the solutions.

For given f ∈ V l−1γ (Ω) with γ ∈ (l + 3/2, l + 5/2), let u, u• be a weak solution according

to Proposition 2.3. Clearly V l−1γ (Ω) ⊂ V l−1

l (Ω) and u ∈ V l+1l (Ω) then. Since A is constant

for |x| > R0 for a suitable radius R0 > 0, we can apply a general result for elliptic boundaryvalues in domains with conical boundary points (see [39, Ch. 6.4], e.g.) to characterize theasymptotic behavior of u at infinity. Thereby, let F denote the fundamental matrix of thedifferential operator L(∇) in R3, i.e.,

L(∇)F (x) = δ(x)I3, x ∈ R3, where L(∇) = D(−∇)>A0D(∇) (2.18)

where δ is the Dirac measure. Since u can be regarded as a V l+1l -solution of an exterior

Dirichlet problem in Ξ = x : |x| > R0, we obtain the asymptotic representation for|x| > R0

u(x) = (d(−∇)F (x)>)> a + (D(−∇)F (x)>)> b + u(x) =: U(x) + u(x) (2.19)

with u ∈ V l+1γ (Ξ)3. The vectors a, b ∈ R6 are coefficient columns, D(ξ) is matrix (1.1)2 and

d(ξ) is a similar matrix (generating the space of rigid motions), defined by

d(ξ)> =

1 0 0 0 αξ3 −αξ2

0 1 0 −αξ3 0 αξ1

0 0 1 αξ2 −αξ1 0

, α =

1√2. (2.20)

If the right hand side f vanishes on Ξ, then u solves the homogeneous system (1.6)1 in Ξ,and due to general results in [24] (see also [39, Ch. 3.6]), the remainder in (2.19) fulfils

|∇kxu(x)| 6 ck (1 + |x|)−3−k , x ∈ Ξ, k ∈ N0. (2.21)

We emphasize that the matrices (1.1)2 and (2.20) satisfy the relations

d(∇)d(x)>∣∣∣x=0

= I6 , d(∇)D(x)>∣∣∣x=0

= O6 ,

D(∇)d(x)> = O6 , D(∇)D(x)> = I6 ,

(2.22)

where ON is the null matrix of size N ×N .

Lemma 2.5 The coefficient column a ∈ R6 in the representation (2.19) is given by theintegral formula

a =∫

Ω•

d(x)f•(x) dx +∫

Ω

d(x)f(x) dx +∫

Γ

d(x)g1(x) dsx .

Proof. Let XR ⊂ C∞0 (R3) be a family of cut-off functions with XR(x) = 1 for|x| 6 R + 1, and XR(x) = 0 for |x| ≥ R + 2. Let u, u• be a weak solution according toDefinition 2.2. We use (2.4) with v = XRdj , where dj is the j-th row of the matrix d, andR ≥ R0 + 2, so that Ω• ⊂ BR−2. With (2.22)3, we obtain

(AD(∇)u,D(∇)(XRdj))Ω = (f•, dj)Ω• + (f,XRdj)Ω + (g1, dj)Γ. (2.23)

Again due to (2.22)3, we have D(∇)(XRdj) = −[D(∇),XR]dj , thus the integrand on the lefthand side of (2.23) vanishes outside the annulus x : R < |x| < R + 2. Moreover we haveXRdj = dj on the sphere SR, while this expression vanishes on SR+2. Thus, integration byparts leads to

(AD(∇)u,D(∇)(XRdj))Ω = (AD(∇)u,D(∇)XRdj)BR+2\BR

= (Nu,XRdj)∂(BR+2\BR) + (Lu,XRdj)BR+2\BR

= (Nu, dj)SR+ (f,XRdj)BR+2\BR

,

8

where N is the Neumann operator defined in (1.7) with n = −R−1x. We have A(x) = A0

for |x| ≥ R0, hence with the associated elasticity and Neumann operators L0 and N 0 weobtain for R ≥ R0

(N 0u, dj)SR= (f•, dj)Ω• + (g1, dj)Γ + (f,XRdj)Ω − (f,XRdj)BR+2\BR

. (2.24)

Since f(1 + r)5/2 ∈ L2(Ω) the integral∫Ω

djf converges and we may pass to the limitR →∞ in the right-hand side of (2.24), note that the last integral in (2.24) vanishes then.To calculate the limit of the left-hand side, we insert the asymptotic representation (2.19)of u into (2.24), then by (2.21), (N u, dj)SR

= O(R−1) as R → ∞, and we are left with theterms

(N 0U, dj)SR,

which can be interpreted as a distribution with compact support applied to the C∞-functiondj . Here we mention that due continuity arguments in spaces of distributions, Green’sformula

(L0u, v)BR− (u,L0v)BR

= (N 0u, v)SR− (u,N 0v)SR

can be extended from u, v ∈ C∞(R3)3 to u = ∂αFk, where Fk is a column of the fundamentalsolution F . Then the first integral has to be replaced by 〈∂αδ, vk〉 = (−1)|α|∂αvk(0). Weapply this argument for u = U , and v = dj , together with formulae (2.22), this leads to

(N 0U, dj)SR〈= d(−∇)>a I3δ, dj〉+ 〈D(−∇)>b I3δ, dj〉

=(d(∇)d>j (x) · a + D(∇)dj(x)> · b)

∣∣∣x=0

= aj .

2

In order to derive an integral formula for the coefficient vector b in the asymptoticrepresentation (2.19) consider problem (1.6) – (1.8) with the special right-hand sides

f•(k)(x) = D(∇)>A•(x)e(k), f(k)(x) = D(∇)>A(x)e(k),

g0(k) = 0, g1

(k)(x) = D(n(x))>(A•(x)−A)e(k),(2.25)

where e(k) is the k-th unit vector in R6. Note that f(k) has a compact support containedin BR0 due to the choice of the matrix A(x) = A0 + Ae(x). The data in (2.25) arise if wereplace u in the transmission problem (1.6), (1.9) by the rows of the matrix −D(x). Wedenote the corresponding unique weak solutions to (1.6) – (1.8) by Z(k) , Z•(k) 3. Since

∫

Ω•d(x)f•(k) (x) dx +

∫

Ω

d(x)f(k) (x) dx +∫

Γ

d(x)g1(k) (x) dsx = 0 ∈ R6 ,

Lemma 2.5 turns the asymptotic form (2.19) for the solution Z(k) into

Z(k)(x) = (D(∇)F (x)>)>P(k) + Z(k)(x) (2.26)

where the remainder Z(k) satisfies (2.21) and P(k)(= −b) denotes a column of height 6. Re-garding Z(k)(x) as a column for each x, we define the 3×6-matrix Z(x) = (Z(1)(x), . . . , Z(6)(x)),and, analogously, Z•(x). Hence, due to (2.25) and (2.22), the columns of the matrix

ζ(x) = D(x)> + Z(x), Z•(x) (2.27)

are formal solutions of the homogeneous problem (1.6), (1.9) (as well as the columns of thematrix d(x)>), although they do not belong to the energy space H. A slight modificationof the proof of Lemma 2.5 (cf. [46, 54]) provides the following assertion.

Lemma 2.6 The coefficient column b ∈ R6 in (2.19) is given by the integral formula

b =∫

Ω•ζ(x)>f•(x) dx +

∫

Ω

ζ(x)>f(x) dx

+∫

Γ

ζ(x)>g1(x) dsx −∫

Γ

D(n(x))>AD(∇)ζ(x)

>g0(x) dsx.

3Note, that the rows of D(x) are solutions to the transmission problem which grow at infinity whileZ(k) , Z•

(k) decays at infinity.

9

2.3 The polarization matrix and its properties.

Rewriting the asymptotic representation (2.26) in the condensed form

Z(x) = (D(∇)F (x)>)>P + Z(x), (2.28)

there appears the matrix P of size 6×6 composed of the coefficient columns P(1), . . . , P(6) in(2.26). As in [36, 46] and others, we call P the polarization matrix for the elastic inclusionΩ•.

By Lemma 2.6 and formula (2.25) we obtain the integral representation

P =−∫

Ω•

(D(x)> + Z•(x)

)>D(∇)>A•(x) dx

−∫

Ω

(D(x)> + Z(x)

)>D(∇)>A(x) dx

−∫

Γ

(D(x)> + Z(x)

)>D(n(x))>

(A•(x)−A(x)

)dsx.

(2.29)

Let us transform the right-hand side of (2.29). Using D(∇)>D(x) = I6 and integratingby parts, we find

∫

Ω•

(A•(x)−A0

)dx +

∫

Ω

Ae(x)dx

=∫

Ω•

(D(∇)D(x)>

)(A•(x)−A0

)dx +

∫

Ω

(D(∇)D(x)>

)Ae(x)dx

= −∫

Ω•D(x)>D(∇)>

(A•(x)−A0

)dx−

∫

Γ

D(x)>D(n(x))>(A• −A0

)dsx

−∫

Ω

D(x)>D(∇)>Ae(x) dx +∫

Γ

D(x)>D(n(x))>Ae(x) dsx

= −∫

Ω•D(x)>D(∇)>A•(x)dx−

∫

Ω

D(x)>D(∇)>A(x)dx

−∫

Γ

D(x)>D(n)>(A•(x)−A(x))dsx,

the last equality holds true due to D(∇)>A0 = 0. Since the columns of Z, Z• are containedin H and fulfill definition 2.2 with data given in (2.25) we may use identity (2.10) withu, u• = Z, Z• = v and obtain further

−∫

Ω•Z•>D(∇)>A• dx−

∫

Ω

Z>D(∇)>Adx−∫

Γ

Z>D(n)>(A−A•

)dsx

= −∫

Ω•

(D(∇)Z•

)>A•D(∇)Z• dx−

∫

Ω

(D(∇)Z

)>AD(∇)Z dx.

Thus, we have another integral representation of the polarization matrix

P = −∫

Ω•

(A0 −A•(x)

)dx +

∫

Ω

Ae(x)dx

−∫

Ω

(D(∇)Z

)>AD(∇)Z dx−

∫

Ω•

(D(∇)Z•

)>A•D(∇)Z• dx.

(2.30)

The last two matrices are but Gram’s matrices for the sets of vector functions Z(k) andZ•(k), hence in particular, they are symmetric and non-negative. Thus we can formulatetwo intrinsic properties of the polarization matrix.

Theorem 2.7 The polarization matrix P is always symmetric. If Ae = 0, i.e. A(x) = A0

everywhere in Ω, and A•(x) < A0 for x ∈ Ω• then P is negative definite.

10

2.4 A homogeneous inclusion.

In this section we assume that the inclusion Ω• as well as the elastic space are homogeneous,i.e., A•, A are constant matrices. We put

Z•(x) = Z∗(x)− Z0(x), Z0(x) = D(x)>(A•)−1(A−A•).

Then the columns of Z,Z∗ satisfy problem (1.6)-(1.8) with

f = 0, f• = 0, g1 = 0, g0(x) = −Z0(k)(x).

Applying Lemma 2.6 to this problem, we derive

−P =∫

Γ

(D(n)>AD(∇)ζ

)>Z0 dsx

=−∫

Γ

(D(n)>AD(∇)(D> + Z)

)>Z dsx

+∫

Γ

(D(n)>A•D(∇)

(D> + Z∗ + Z0

))>Z∗dsx

=−∫

Γ

(D(n)>AD(∇)Z

)>Z dsx +

∫

Γ

(D(n)>A•D(∇)Z∗

)>Z∗dsx

−∫

Γ

(D(n)>A

)>Z dsx +

∫

Γ

(D(n)>A•

(I6 + (A•)−1(A−A•)

))>Z∗ dsx. (2.31)

The sum of the first two integrals in the right-hand side of (2.31) is equal to

−∫

Ω

(D(∇)Z

)>AD(∇)Z dx−

∫

Ω•

(D(∇)Z∗

)>A•D(∇)Z∗ dx

and gives rise to a non-positive symmetric 6 × 6-matrix. The sum of the last two integralsin (2.31) coincides with

∫

Γ

(D(n(x))>A

)>Z0(x) dsx =

∫

Ω•

AD(∇)D(x)>(A•)−1(A−A•) dx

= A (A•)−1 (A−A•)∣∣Ω•∣∣ = A

[(A•)−1 −A−1

]A

∣∣Ω•∣∣, (2.32)

where∣∣Ω•

∣∣ denotes the volume of the domain Ω•. Thus we have proved the followingassertion.

Theorem 2.8 If the matrix A• is constant and (A•)−1 < A−1, then the polarization matrixP is positive definite.

Certain positivity/negativity properties of the polarization matrix P can be expressed interms of the eigenvalues λ1, . . . , λ6 of the matrix A−1/2A•A−1/2. This matrix is symmetricand positive definite, and hence λj > 0 and the eigenvectors aj ∈ R6 can be normalized bythe condition

(ak

)>aj = δj,k , j, k = 1, . . . , 6. Then the columns bj = A−1/2aj satisfy the

formulaeA•bj = λjAbj ,

(bk

)>Abj = δk,j . (2.33)

Theorem 2.9 1) If λj > 1 then(bj

)>Pbj > 0.

2) If λj < 1 then(bj

)>Pbj < 0.

3) If λj = 1 then(bj

)>Pbj = 0.

11

Proof. 1) Recalling (2.31) – (2.32), we see that −P 6(A(A•)−1 A − A

) ∣∣Ω•∣∣. Thus, invirtue of (2.33),

−(bj

)>Pbj 6

(bj

)>(A(A•)−1A−A

)bj

=(bj

)>A(λ−1

j − 1)bj∣∣Ω•∣∣ = (λ−1

j − 1)∣∣Ω•∣∣ < 0. (2.34)

2) By (2.30), we have P 6 (A• −A) |Ω•| and

(bj

)>Pbj 6

(bj

)>(A• −A

)bj

∣∣Ω•∣∣ =(bj

)>A(λj − 1)bj

∣∣Ω•∣∣ = (λj − 1)∣∣Ω•∣∣ < 0.

(2.35)3) Repeating calculations (2.34) and (2.35), we change “< 0” for “= 0” to see the

assertion. 2

Remark 2.10 (i) Relations to the elastic moment tensors used in [3]. For isotropic ho-mogeneous inclusions in a homogeneous and isotropic elastic space, in [7] and [3, Ch. 7.2]so called first order EMTs where considered which belong to the representation formula(2.36) below. If e(j) denotes the jth unit vector in R3, then there exist solutions to thehomogeneous transmission problem (1.6), (1.8) in the form

hij = xie(j) +3∑

p,q=1

mijpq∂pFq + O(|x|−3), as |x| → ∞, (2.36)

here Fq is again the qth column of the fundamental solution matrix. Let us denote thecolumns of d(x)> (see (2.20)) and D(x)> (see ((1.1)) by dk and Dk, respectively. Using therelations

xie(i) = Di, i = 1, 2, 3,

x2e(3) = α(D4 + d4), x3e(1) = α(D5 + d5), x1e(2) = α(D6 + d6)x3e(2) = α(D4 − d4), x1e(3) = α(D5 − d5), x2e(1) = α(D6 − d6),

elementary calculations show, that the tensor mijpq is related to the polarization matrix in

exactly the same manner like the Hooke tensor to the stiffness matrix while using the Voigtnotation, in particular the formula (1.4) holds true with A replaced by P and cij

pq by mijpq.

From the arguments of section 2 it is clear that the tensor mijpq is well defined also under the

more general assumptions of Section 1.1, moreover it is always symmetric, and the positivityor negativity results [3, Sect. 7.2] are a particular case of Theorem 2.7 and Theorem 2.8.

(ii) Polarization matrices of higher order. We further emphasize that a straightforwardargumentation leads to polarization matrices of arbitrary order, even in the inhomogeneouscase, provided Ae in (1.5) has compact suppport (see e.g. [52, 45]). Let us introduce the no-tation L0-harmonic for a solution h to the problem L0h = 0 in R3, where L0 is the elasticityoperator related to the constant stiffness matrix A0 in R3. Instead of the Taylor expansionof h around zero it is more convenient to use an expansion into homogeneous L0-harmonicpolynomials. Let U1, . . . UK denote a basis of homogeneous L0-harmonic polynomial vec-tor fields generating the L0 harmonic fields (h1, h2, h3)>, where each hj is a polynomialup to order k. It is always possible to fix such a basis by the normalization conditionUk(∇)>U j(x)|x=0 = δkj (recall that we read the fields U j as columns), thereby U1, . . . , U12

can be found as the columns of d> and D>. If we put V k = Uk(∇)>F then any derivative∂αFk, |α| 6 k, can be uniquely represented in terms of certain V j , cf. [52, p. 242]. Thenwe find solutions to the homogenous transmission problem in the form

ζi(x) = U i(x) +K∑

j=1

PijVj(x) + O(|x|−2−k) as |x| → ∞.

Since we included the columns of d(x), we have Pij = Pji = 0 for i = 1, . . . , 6. The(K − 6)× (K − 6) matrix P = (Pij)K

i,j=7 is the polarization matrix of order k.

12

2.5 Affine transformations

Again we consider the case of homogeneous solids, i.e. A and A• are constant matrices. Inaddition, we assume that the inclusion Ω•h is defined through rescaling:

Ω•h = x : h−1x ∈ Ω• , (2.37)

where h > 0 is the similarity coefficient. Since the matrix function D(x) depends linearlyon the variables xj , the solution ζh of the homogeneous elasticity problem for the compositeelastic space Ωh ∪ Ω•h is related to the solution ζ by the formula

ζh(x) = h−1ζ(hx) .

Thus, in view of formula (2.28), we obtain that

P (Ω•h) = h−3P (Ω•) .

In [20, 21] the affine transformation

x 7→ x = tx (2.38)

is applied to the elasticity problem in three spatial dimensions and the notion of algebraicallyequivalent anisotropic media (see [1, 18]) is used to evaluate the fundamental matrix solu-tion F (x) = (t>)−1F (t−1x)t−1. Thereby, t = (tjk)3j,k=1 is an arbitrary, not necessarilyorthogonal, matrix of size 3× 3. In view of the coordinate dilatation (2.37) we may assumethat

det(t) = 1 ,

i.e., the affine transformation preserves volume. As verified in [20, 21] matrices A,A• aretransformed in the composite plane Ω ∪ Ω• = tΩ ∪ tΩ• into the new Hooke’s matrices

A = TAT> , A• = TA•T> ,

where the (6× 6)-matrix T depends on the entries of t in the following way

T =

t211 t212 t213√

2 t12t13√

2 t11t13√

2 t11t12

t221 t222 t223√

2 t22t23√

2 t21t23√

2 t21t22

t231 t232 t233√

2 t32t33√

2 t31t33√

2 t31t32√2 t21t31

√2 t22t32

√2 t23t33 t23t32 + t22t33 t23t31 + t21t33 t22t31 + t21t32√

2 t11t31√

2 t12t32√

2 t13t33 t13t32 + t12t33 t13t31 + t11t33 t12t31 + t11t32√2 t11t21

√2 t12t22

√2 t13t23 t13t22 + t12t23 t13t21 + t11t23 t12t21 + t11t22 .

.

From the calculations in [21] it follows that the polarization matrix P (Ω•) for the alge-braically equivalent composite plane Ω ∪ Ω• takes the form

P (Ω•) = TP (Ω•)T> . (2.39)

Formula (2.39) becomes explicit if the original matrix P (Ω•) is known. If Ω•

is a cavity,then affine transformations of type (2.38) are useful to simplify the anisotropic propertiesof the medium. For example, in two spatial dimensions it is known that any anisotropicmaterial is algebraically equivalent to an orthotropic material with four symmetry axes (see[1, 18]).

3 Shape sensitivity analysis for polarization matrices

The direct approach of the shape sensitivity analysis is used in two spatial dimensions. Thereis no major difficulty to apply the same approach in the three spatial dimensions, howeverwe prefer to perform such an analysis in the framework of standard boundary variationtechnique with material derivatives and shape gradients of energy functionals. In this waywe show that two general approaches of the shape sensitivity analysis are applicable forpolarization matrices.

13

3.1 Two dimensional problems

We start with the same assumptions as in Section 2.4, only as a two-dimensional problem,i.e. R2 = Ω∪Ω•, where Ω• ⊂ R2 is bounded by a smooth contour Γ, and the elastic materialswhich fill up Ω• and Ω = R2Ω• are homogeneous with constant Hooke’s matrices of size3 × 3 (see Section 1.1). The existence of a weak solution with finite energy can be provedexactly as in Section 2,

In a neighborhood U of Γ we introduce a curvilinear coordinate system (s, n), where nis the oriented distance to Γ, n > 0 in Ω•, and s is the arc length on Γ. In our notationa point on Γ is identified with its coordinate s. Moreover, a function x 7→ v(x), after thechange of variables is still denoted v(n, s). By an appropriate rescaling, the diameter of theinclusion becomes 1, so that all coordinates are dimensionless. Given a small parameter hand a function H ∈ C2(Γ), we introduce the perturbed contour

Γh = x ∈ U : s ∈ Γ , n = hH(s) , (3.1)

it becomes the boundary of the perturbed inclusion Ω•h, while Ωh = R2Ω•h. We use thenotation introduced in Sections 2.1 and 1.1. The polarization matrix P (h) of size 3× 3 forthe inclusion Ω•h is determined from the problem

LZh(k)(x) = 0 , x ∈ Ωh (3.2)

L•Zh•(k)(x) = 0, x ∈ Ω•h (3.3)

NZh(k)(x)−N •Zh•

(k)(x))

= −D(nh(x))>(A−A•)ek,

x ∈ Γh , (3.4)

Zh(k)(x) = Zh•

(k)(x) , x ∈ Γh (3.5)

where ek = (δ1,k, δ2,k, δ3,k). The solutions Zh(k), Z

h•(k) exist and the (2 × 3)-matrix Zh =

(Zh1, Zh2, Zh3) admits the asymptotic form

Zh(x) = (D(∇)F (x)>)>P (h) + O(|x|−2). (3.6)

We are going to find the second term in the asymptotics of the polarization matrix

P (h) = P + hP + . . . (3.7)

It is reasonable to suppose that the solutions Zk, Z•k and the matrix P are known forthe unperturbed inclusion Ω• = Ω•0. Then, we need to find the correction terms in theexpansions

Zh(k)(x) = Zk(x) + Zk(x) + . . . (3.8)

Zh•(k)(x) = Z•k(x) + Z•k(x) + . . .

According to (2.27) the matrix functions ζh = D> + Zh and ζ = D> + Z solve the homo-geneous problem (3.2)-(3.5) in Ωh ∪ Ω•h and Ω ∪ Ω•, respectively. That is why, it is moreconvenient to consider the expansions of theirs columns

ζh(k)(x) = ζk(x) + Zk(x) + . . . ,

ζh•(k)(x) = ζ•k(x) + Z•k(x) + . . . ,

which immediately follow from (3.8).Since the boundary Γ is smooth, the vector functions ζk and ζ•k can be extended in the

Sobolev class H2 from Ω and Ω• onto Ω ∪ U and Ω• ∪ U , respectively. The extensions arestill denoted by ζk and ζ•k. Clearly, Zk and Z•k have to verify the homogeneous elasticitysystem

D(−∇)>AD(∇)Zk(x) = 0 , x ∈ Ω , (3.9)D(−∇)>A•D(∇)Z•k(x) = 0 , x ∈ Ω• . (3.10)

14

It remains to evaluate the main parts of discrepancies in the homogeneous transmissionconditions (3.4) and (3.5) which appear due to shift (3.1) of the contact contour.

We introduce the projections ζn and ζs of the displacement vector ζ = (ζ1, ζ2) on theaxes n and s, respectively,

ζn(n, s) = n1(s)ζ1(n, s) + n2(s)ζ2(n, s) ,

ζs(n, s) = −n2(s)ζ1(n, s) + n1(s)ζ2(n, s) ,

where n = (n1, n2)> stands for the unit normal vector on Γ interior to the inclusion Ω•.The curvilinear components of the strain tensor ε(ζ) take the form

εnn(ζ) = n21ε11(ζ) + n2

2ε22(ζ) + 2n1n2ε12(ζ) ,

εss(ζ) = n22ε11(ζ) + n2

1ε22(ζ)− 2n1n2ε12(ζ) ,

εns(ζ) = εsn(ζ) = n1n2(ε11(ζ)− ε22(ζ)) + (n21 − n2

2)ε12(ζ) .

In terms of displacements the components of the strain tensor are expressed as follows:

εnn(ζ) = ∂nζn , εss(ζ) = J−1(∂sζs + κζn) , (3.11)

εns(ζ) = εsn(ζ) =12(∂nζs + J−1(∂sζn − κζs)) ,

where ∂n = ∂/∂n, ∂s = ∂/∂s, J(n, s) = 1 + nκ(s) is the Jacobian and κ(s) is the curvatureof the arc Γ at the point s.

The Hooke’s law (1.3) in the coordinates (n, s) takes the form

σnn(ζ;n, s)σss(ζ; n, s)√2σns(ζ; n, s)

= A(s)

εnn(ζ;n, s)εss(ζ; n, s)√2εns(ζ;n, s)

, A(s) = Θ(s)>AΘ(s) ,

where the matrix Θ(s) is given by the formula

Θ =

cos2 ϑ sin2 ϑ −2−1/2 sin 2ϑsin2 ϑ cos2 ϑ 2−1/2 sin 2ϑ

2−1/2 sin 2ϑ −2−1/2 sin 2ϑ cos 2ϑ

with cosϑ = n1 and sinϑ = n2.For the transmission condition (3.5) the Taylor formula

ζ(hH(s), s) = ζ(0, s) + hH(s)∂nζ(0, s) + O(h2)

readily yields

Zk(0, s)−Z•k(0, s) = H(s)(∂nζk(0, s)− ∂nζ•k(0, s)) , s ∈ Γ . (3.12)

Since (∂n, J(n, s)−1∂s) is the gradient operator in the curvilinear coordinates, the unitnormal nh and the tangential vector sh on Γh have the components

nhn(s) = sh

s (s) = jh(s)−1/2, nhs (s) = −sh

n(s) = −hjh(s)−1/2∂sH(s),

where jh(s) stands for the normalizing factor 1 + h2H ′(s)2 and H ′ = ∂sH. Hence

σnhnh(ζ + hZ;hH(s), s) = σnn(ζ; 0, s) + h(H(s)∂nσnn(ζ; 0, s)

−2H ′(s)σns(ζ; 0, s) + σnn(Z; 0, s)) + O(h2) ,

σnhsh(ζ + hZ;hH(s), s) = σns(ζ; 0, s) + h(H(s)∂nσns(ζ; 0, s)

+H ′(s)(σnn(ζ; 0, s)− σss(ζ; 0, s)) + σns(Z; 0, s)) + O(h2) .

(3.13)

15

On the other hand, the equilibrium equations in the coordinates (n, s) take the form

−∂nσnn(ζ)− J−1(∂sσns(ζ) + κ(σnn(ζ)− σss(ζ))) = 0 ,

−∂nσns(ζ)− J−1(∂sσss(ζ) + 2κσns(ζ)) = 0

and for n = 0 we obtain

∂nσnn(ζ; 0, s) = −∂sσns(ζ; 0, s)− κ(σnn(ζ; 0, s)− σss(ζ; 0, s)) , (3.14)∂nσns(ζ; 0, s) = −∂sσss(ζ; 0, s)− 2κσns(ζ; 0, s)) .

Now, we are able to conclude on the second transmission condition for Zk,Z•k. Sinceζk, ζ•k satisfy the homogeneous transmission conditions on the unperturbed contour Γ, thefollowing identities are still valid on Γ

ζkn = ζ•kn , ζk

s = ζ•ks , σnn(ζk) = σnn(ζ•k) , σns(ζk) = σns(ζ•k) , (3.15)

this considerably simplifies the above formulae. In particular, taking into account the secondrelations in (3.13), (3.14) we have

σns(Zk)− σ•ns(Z•k) = −H(∂nσns(ζk)− ∂nσ•ns(ζ

•k))−H ′ (σnn(ζk)− σ•nn(ζ•k))

+H ′ (σss(ζk)− σ•ss(ζ•k)

)= H

(∂sσss(ζk)− ∂sσ

•ss(ζ

•k) + 2κ(σns(ζk)− σ•ns(ζ•k)))

+H ′ (σss(ζk)− σ•ss(ζ•k)

)= ∂sH

(σss(ζk)− σ•ss(ζ

•k))

,

note that the terms in vanish due to (3.15). Similar calculations, in view of the firstrelations in (3.13), (3.14) lead to the couple of transmition conditions

σnn(Zk; 0, s)− σnn(Z•k; 0, s) = −κ(s)H(s)(σss(ζk; 0, s)− σss(ζ•k; 0, s)) , (3.16)

σns(Zk; 0, s)− σns(Z•k; 0, s) = ∂s(H(s)(σss(ζk; 0, s)− σss(ζ•k; 0, s)) , s ∈ Γ .

Problems (3.9), (3.10), (3.12), (3.16) with k = 1, 2, 3 admit decaying matrix solutions

Z = (Z1,Z2,Z3) , Z• = (Z•1,Z•2,Z•3)of the form

Z(x) = (D(∇)F (x)>)>P + O(|x|−2) (3.17)

(cf. (3.6)). According to Lemma 2.6, the entries of (3×3)-matrix P in (3.17) and (3.7) havethe integral representation

Pjk =∫

Γ

(ζj)>(σ(n)(Zk)− σ•(n)(Z•k))dsx −∫

Γ

(Zk −Z•k)>σ(n)(ζj)dsx =: I1 − I2 .

(3.18)

Note, that in view of the homogeneous conditions (3.5) and (3.4), the identities ζj = ζ•j

and σ(n)(ζj) = σ•(n)(ζ•j) hold true on Γ, which has been taken into account in (3.18).Using formulae (3.11) and (3.15) it follows that

∂nζk − ∂nζ•k = 2εns(ζk)− 2εns(ζ•k) on Γ

and, by (3.12),

I2 = −∫

Γ

(εnn(ζk)− εnn(ζ•k))σnn(ζj) + 2(εns(ζk)− εns(ζ•k))σns(ζj)dsx .

Furthermore, applying (3.16), (3.15), (3.11) and integrating by parts on the contour Γ resultin

I1 =∫

Γ

(ζjs∂s(H(σss(ζk)− σ•ss(ζ

•k)))− ζjnκH(σss(ζk)− σ•ss(ζ

•k)))dsx

−∫

Γ

H(σss(ζk)− σ•ss(ζ•k))(∂sζ

js + κζj

n)dsx −∫

Γ

H(σss(ζk)− σ•ss(ζ•k))εss(ζj)dsx .

(3.19)

16

In order to explain the physical sense of the obtained expression, we recall the notion of thesurface enthalpy [51, 41], which is common in many applications in mechanics, namely, theintegral

Ξ(ζ) =12

∫

Γ

ξ(ζ, ζ)dsx (3.20)

with the density4

ξ(ζ, η) = σnn(ζ)εnn(η) + σns(ζ)εns(η) + σsn(ζ)εsn(η)− εss(ζ)σss(η) .

We emphasize, that all terms evaluated for ζ and ζ• are continuous on Γ in view of thehomogeneous transmission conditions (3.5), (3.4), while all terms evaluated for η may havejumps over Γ.

¿From (3.18) - (3.19) it is straightforward to obtain

Pjk =∫

Γ

H(s)ξ(ζj , ζk; s)dsx −∫

Γ

H(s)ξ•(ζ•j , ζ•k; s)dsx . (3.21)

In other words, the matrix P in (3.7) is given by the jumps of the weighted enthalpy obtainedfor the special solutions ζ of the homogeneous problem (3.2)- (3.5) for the composite planeΩ ∪ Ω• with the original contact contour Γ.

We have performed the formal asymptotic analysis which can be justified in the sameway as in [51] for elasticity problems in domains with rapidly oscillating boundaries (butwith the serious simplifications, see also [40] for a phase transition problem in the sameframework).

Proposition 3.1 The polarization matrix P (h) of the composite plane Ωh ∪ Ω•h with theperturbed contact contour (3.1) has the asymptotic form (3.7) where the entries of matrix Pare provided in (3.21), and the remainder is of order O(h2).

Let us consider the elastic plane with the hole Ω•h bounded by the perturbed contour(3.1), this means the transmission problem (1.6), (1.8) changes to the Neumann problem

Lu = f,Nu = g on Γ. (3.22)

Then, ζj(x) = Dj(x) + Zj(x) are solutions to the homogeneous exterior elasticity problem(3.22) with linear growth in infinity, and the representation formula (3.6) holds as well. Theformula (3.7) in Proposition 3.1 crucially simplifies, namely

Pjk =−∫

Γ

H(s)εss(ζj ; 0, s)εss(ζk; 0, s)ds

=−∫

Γ

H(s)(A−1(s)

)22

σss(ζj ; 0, s)σss(ζk; 0, s)ds ,

(3.23)

since A• = 0 and σnn(ζp) = σns(ζp) = 0 on Γ and, moreover, inverting the Hook’s law(3.11) we obtain

εss(ζj ; 0, s) =(A(s)−1

)21

σnn(ζp; 0, s)+

+(A(s)−1

)22

σss(ζj ; 0, s) +√

2(A(s)−1

)23

σns(ζj ; 0, s)

=(A(s)−1

)22

σss(ζj ; 0, s), s ∈ Γ .

The entry(A(s)−1

)22

of the compliance matrix A(s)−1 is positive, and, hence, the matrixP with elements (3.23) is positive definite in the case when the hole enlarges, that is forH(s) < 0 (recall that n is the outward normal with respect to Ω = R2Ω•). In contrast,shrinking the hole leads to a decrease of eigenvalues of the polarization matrix P .

Note that completely analoguous properties hold true for the virtual mass tensor (Polya-Szego tensor) related to the exterior Neumann problem for the Laplacian.

4The quadratic functional (3.20) is but a Gibbs functional, obtained from the surface energy functionalby the partial Lagrange transformation for the tangential components of the stress/strain state [].

17

3.2 Shape sensitivity analysis in three spatial dimensions

The classical shape sensitivity analysis in smooth domains can be applied in order to derivethe shape derivatives of the polarization matrix. In the special case g0 = 0 in (1.9) the weaksolution to the transmission problem (see Definition 2.2) minimizes the energy functional

J0(Γ) =12(AD(∇)u), D(∇)u)Ω +

12(A•D(∇)u•, D(∇)u•)Ω• (3.1)

−(f, u)Ω − (f•, u•)Ω• − 〈g1, u = u•〉Γ =:12a(u, u)− L(u).

Recalling the representation (2.30) we observe that the entries of the polarization matrixcan be obtained using such functionals which simplifies the derivation. Thus we start withan auxiliary result on material derivatives of the minimizers to (3.1), which seems to beinteresting on its own. We point out that, in view of the representation (3.27) of thepolarization matrix, the shape derivatives of the polarization matrix P with respect to theperturbations of the interface Γ can be deduced from the shape differentiability of the energyfunctional (3.1), the resulting shape derivatives are given by formula (3.28). The existenceof the shape derivative dJ(Γ; V ) of the functional (3.1) follows by Lemma 3.4 combined withthe standard results on the differentiability of volume and surface i ntegrals [57] which arelisted below, in Lemma 3.2. The result is established by the differentiability of the functional(3.25) with respect to the parameter t at t = 0. Another problem, important for numericalapplications of our results, is the identification of the obtained shape gradient gΓ given informula (3.26). The actual form of the shape gradient can be given by the derivation withrespect to t in the variable domain setting of our problem. We exploit to this end the shapedifferentiability of the energy type shape functionals, taking into account the decompositionof P in (3.21), (3.22) combined with the representations (3.23), (3.24) for the entries of thefunctionals in (3.1). All steps in this derivation of formula (3.28) can be performed alongthe lines of [57].

3.3 Deformations of interfaces

Let Ξ ⊂ R3 be a domain with smooth compact boundary Γ of class C2,α at least, we canthink of Ξ ∈ Ω, Ω•, and V ∈ Ck

0 (R3)3, k ≥ 1, be a fixed given mapping with compactsupport in an open neighbourhood of Γ. To V we relate the family of diffeomorphismsTt : R3 7→ R3 with inverse T−1

t defined by

Tt(x) := x + tV (x) = (I + tV )(x) , t ∈ [0, ε0).

We denote the change of variables by

y(x) := Tt(x),

and also introduce the families of transported domains and interfaces

Ξt := Tt(Ξ), Γt := Tt(Γ). (3.2)

If g(t, ·) := gt ∈ H1(Ξt) for t ∈ [0, ε0), then

gt := gt Tt ∈ H1(Ξ) (3.3)

and with g = g(0, ·) ∈ H1(Ξ) the strong (weak) material derivative is defined as

g(V ) = limt→0

1t(gt Tt − g) = lim

t→0

1t(gt − g)

provided this limit exists strongly (weakly) in H1(Ξ). Thereby the Sobolev space H1(Ξ) canbe replaced by any other Sobolev space (eventually under additional regularity assumptionsfor V ). In a similar manner, we can define g for functions defined on Γt. If the weak

18

material derivative exists in H1(Ξ), then the shape derivative g′ ∈ L2(Ξ) in the direction ofV is defined by

g′ = g −∇g · V (3.4)

If g ∈ H2(Ξ), then the tangential gradient ∇Γg := ∇xg − (∇xg · n)n exists in H1/2(Γ) andwe can define the boundary shape derivative

g′Γ = g −∇Γg · V,

again provided g exists in H1/2(Γ) at least as a weak limit. In the following we deal withtwo kinds of functions defined in the domain Ξt. The first type are functions defined inthe whole space R3 and then restricted to Ξt. They appear as data for our problems, andtheir so-called shape and material derivatives are explicitly given [57]. The second typeof functions are the solutions to boundary value problems defined in the variable domains,especially solutions to the transmission problem. For these functions the material derivativesare defined by solutions of auxiliary boundary value problems defined in the fixed domainΞ, and there is a relation between shape and material derivatives [57]. We use the followingnotation for the functions and variables

y := y(x) = Tt(x) ∈ Ξt,

Tt(x) := ∇xTt(x) = (∂yi/∂xj)i,j=1,2,3,

ϑt(x) := det(Tt), T−>t := (T−1t )>.

Note that ϑt > 0 if t is small enough. Clearly we have

∂tTt(x)|t=0 = ∇xV (x) = T′(x),∂tϑt(x)|t=0 = ∇x · V (x) = ϑ′(x),

∂tT−>t (x)|t=0 = −(∇V (x))> = (T−>)′(x),

the last formula follows if we observe that T|t=0 coincides with the unit matrix I 5. We recallsome results on the transport of differential operators and integrals, they are used in orderto obtain the derivatives with respect to the parameter t at t = 0 of volume and surfaceintegrals. For the proofs we refer to [57].

Lemma 3.2 For Ξ, Ξt, and gt as in (3.2) and (3.3), respectively, we have∫

Ξt

gt(y)dy =∫

Ξ

gt(x)ϑt(x)dx,

∫

Γt

gt(y)dsy =∫

Γ

gt(x)θt(x)dsx, (3.5)

where θt(x) is the surface Jacobian, i.e. θt(x) = |ϑt(x) (T−1t (x))> · n(x)|, n(x) the unit

normal vector in x ∈ Γ. The derivatives of the integrals (3.5) for t = 0 take the form

d

dt

∫

Ξt

gt(y)dy|t=0 =∫

Ξ

[g(x) + g(x)ϑ′(x)]dx,

d

dt

∫

Γt

gt(y)dsy|t=0 =∫

Γ

[g(x) + g(x)θ′(x)]dsx,

where ϑ′(x) = ∇ · V (x) and θ′(x) = ∇ · V (x)− n(x)>∇V (x)>n(x).

Next we need to introduce transported differential operators. Let gt ∈ Hk(Ξt), if we thinkof ∇ygt as a column vector, then the chain rule for derivatives gives

(∇ygt(Tt(x)) = T−>t (x) · ∇xgt := Nt(x)gt, (3.6)

where Nt(x) can be considered as the transported ∇-operator. Eventually diminishing theinterval [0, ε0), we may assume

KV ε0 =: ‖∇xV ; L∞(R3)‖ ε0 < 1. (3.7)5The notation T′(x) etc is consistent with (3.4), it we observe that for a sufficiently regular regular

function g(t, x) defined on an open neighbourhood of⋃

t∈[0,ε0) ⊂ R4 it follows: g′(x) = ∂tg(t, x)|t=0, and

g(x) = ddt

g(t, Tt(x))|t=0

19

Since T = I+ t∇xV , where I is the 3× 3 unit matrix, we have

T−>t (x) =∞∑

ν=0

(−1)ktk(∇xV (x)>)k, |t| 6 ε0 (3.8)

Interchanging differentiation and summation, elementary calculations show

‖T−>t ; L∞‖ 6 11− ε0KV

, (3.9)

‖∇xT−>t ; L∞‖ 6 1(1− ε0KV )2

‖∇V ; L∞‖, (3.10)

‖∇2xT−>t ; L∞‖ 6 C

( ‖∇V ; L∞‖2(1− ε0KV )3

+‖∇2V ; L∞‖(1− ε0KV )2

), (3.11)

here C is an absolut constant independent of V and t. The relations (3.6) and (3.8) imply

∇ygt Tt =(I− t(∇xV )> + t2((∇xV )>)2T−>t

) · ∇xgt. (3.12)

3.4 The transported transmission problem

Now we return back to our transmission problem (1.6) – (1.8). Let Ω, Ω• and Γ be definedas in Section 2.1, again Γ of class C2,α at least. We fix V and Tt as as in Section 3.3, thenΓt = TtΓ is of class C2,α for all t ∈ [0, ε0),. In addition to the assumptions of Section 2.1 onthe Hooke matrices A•(x), A(x) = A0 +Ae(x), we require now A•, Ae ∈ C1

0 (R3), and A•(x)is positive definit on a compact subset K ⊂ R3, which contains

⋃t∈[0,ε0)

Ω•t . To the matrixdifferential operator D(∇y) we associate the transported operator Dt(x,∇x) =: D(Nt(x)).With

At = A Tt, A•,t = A• Tt,

we define the transported elasticity operators on Ω, Ω• by

L(•),t(x,∇x) = −Dt(x,∇x)>A(•),tDt(x,∇x). (3.13)

Note that Lt, L•,t are second order differential operators with variable coefficients which canbe applied to any u, u• ∈ V 2

β (Ω)×H2(Ω•). In a similar manner, we introduce transportedboundary operators. By Nt and N •

t , we denote the Neumann operators associated with theproblem (1.6) – (1.8) on Ωt, Ω•t . For x ∈ Γ, y = Ttx ∈ Γt, let n(x) and nt(y) be unit normalvectors in x and y (with the same orientation) to Γ and Γt, respectively. Then it holds (see[57, Sect. 2.17])

nt(x) := nt(Tt(x)) =∣∣T−>t · n(x)

∣∣−1T−>t · n(x), (3.14)

and we setN (•)

t = D>(nt(x)) ·A(•),t(x)Dt(x,∇x). (3.15)

The particular representations (3.8) and (3.12) enables us to control the behavior of theoperators (3.13) and (3.15) at t = 0. To formulate our results, we introduce the followingnatural spaces for the data and for strong solutions to our transmission problems:

Dt = V 21 (Ωt)3 ×H2(Ω•t )

3,

Rt = V 01 (Ω)3 × L2(Ω•)3 ×H3/2(Γt)3 ×H1/2(Γt)3,

and the operators related to the transmission problems

At : Dt → Rt, (u, u•) 7→ (Lu,L•u•, (u− u•)|Γt , (Ntu−N •t u•)|Γt).

To the transported operators (3.13) and (3.15) we relate the problem operator At. For t = 0,we simply write D , R and A .

Lemma 3.3 Let V , Tt be defined as in Section 3.3, and t ∈ [0, ε0], so that Tt is a diffeo-morphism and (3.7) holds.

20

(i) The mapping It : (u, u•) → (ut, u•t ) = (u Tt, u

• Tt) defines a bounded isomorphismbetween Dt and D . The norm of It as well as the norm of the inverse It is boundedindependent of t. An analogous result holds for the spaces Rt and R.

(ii) To each set of data Ft = (ft, f•t , g0,t, g1,t) ∈ Rt there exists a unique solution Ut =

(ut, u•t ) ∈ Dt to the problem AtUt = Ft, and

‖Ut; Dt‖ 6 Ct‖Ft; Rt‖.

(iii) If AtUt = Ft, where Ut ∈ Dt, and Ft ∈ Rt, then by (i), U t ∈ D , F t ∈ R, andA tU t = F t, in particular for t ∈ [0, ε0), the mapping A t is a bounded isomorphismfrom D to R.

(iv) Let L (D , R) denote the space of bounded linear operators from D to R, provided withthe usual operator norm. The mapping t 7→ A t is continuous from [0, ε0) to L (D , R),in particular we have limt↓0 A t = A .

Proof. The first assertion follows from the representation (3.6), the estimates (3.9) – (3.11)and the transformation formula (3.5). Part (ii) follows from Proposition 2.3 together withlocal regularity results [56] and [52, Sect. 4.3].

For the proof of (iii) we observe that the definition of the transformed operators andfunctions immediately imply the equation A tU t = F t. The isomorphism property followsfrom part (i) and (ii) since A t = It At It, where the mappings It : D → Dt andIt : Rt → R are defined by ItU = (u T−1

t , u• T−1t ) and ItFt = F t.

Turning to (iv), we first prove the continuity result. Obviously the mapping t → Tt

is in C∞([0, τ ]; C30 (R3)) even for any large τ > 0. Since A• ∈ C1(K), we obtain then for

t, t0 ∈ [0, ε0)

‖A•,t0 −A•,t; L∞(K)‖+ ‖∇(A•,t0 −A•,t); L∞(K)‖ → 0, (3.16)

similarly, since A(x) = A0 + Ae(x)

‖At0 −At;L∞(R3)‖+ ‖∇(At0 −At); L∞(R3)‖ → 0, as t → t0. (3.17)

The representation (3.12) gives

Dt(x,∇x) = D(∇x)− tD((∇xV )>∇x) + t2D(t, x,∇x),

D(t, x,∇x) =∞∑

ν=2

(−1)νtν−2D(((∇V )>)k∇x).(3.18)

This means that D(t, x,∇x) is a first order (matrix) differential operator, where the coeffi-cient functions are bounded in C2(R3) uniformly in t and vanish for large x, say |x| > R0.Representation (3.18) implies that the mapping t → Dt ∈ L (V k

β (Ω), V k−1β (Ω)) as well as

t → Dt ∈ L (Hk(Ω•), Hk−1(Ω•)) for k = 1, 2, β ∈ R, is even Frechet-differentiable for allt ∈ [0, ε0), where the derivative is just the operator D((∇xV )>∇) acting between the corre-sponding Sobolev spaces. Furthermore, representation (3.14) shows that again for small t,the vector fields nt, ∂tnt ∈ C2(Γ) and depend continuously on t. From here the continuityassertion follows using (3.16), (3.17). 2

We now assume that the data for the problem on the t-dependent domains are generatedby restrictions of fixed vector fields, then Lemma 3.3 leads to the following result.

Corollary 3.4 For given

f• ∈ L2(R3)3, f ∈ V 01 (R3)3, G0 ∈ H2(R3)3, G1 ∈ H1(R3)3 (3.19)

we set

ft = f |Ωt , f•t = f•|Ω•t , g0,t = G0|Γt , g1,t = G1|Γt . (3.20)

21

Let Ut := ut, u•t and U = u, u• be the corresponding solution to problem (1.6) – (1.8)

on Ωt, Ω•t , Γt, and Ω, Ω•, Γ, respectively, further U t := ut, u•,t denote the transportedvector field according to (3.3). Then limt↓0 U t = U strongly in D , i.e.

‖ut − u; V 21 (Ω)‖+ ‖u•,t − u•;H2(Ω)‖ → 0, as t → 0.

Proof. With F t defined by the data (3.20) we have F t → F = F0 strongly in R. Part (iv)of Lemma 3.3 implies A t,−1 converges to A −1 in L (R, D), hence

‖U t − U ;D‖ = ‖A t,−1Ft −A −1F ;D‖6 ‖A t,−1F t −A t,−1F ; D‖+ ‖A t,−1F −A −1F ; D‖ → 0. 2

On the other hand, the material derivatives are not really needed for the shape deriva-tives of the polarization tensor. Indeed, representation (2.30) shows that each entry of thepolarization tensor is composed from two kinds of functionals. Recall that

Z(k) := Z(k), Z•(k), k = 1, . . . , 6,

is the solution to the transmission problem with data (2.25). The entries Pkl of the polar-ization tensor can be calculated as follows

Pkl = Pkl + 2a(Z(k),Z(l)), where (3.21)

Pkl : = −∫

Ω•

(A0

kl −A•kl(x))

dx +∫

Ω

Aekl(x) (3.22)

a(Z(k), Z(l)) : = −12(A(D(∇)Z(k), D(∇)Z(l))Ω

− 12(A•(D(∇)Z•(k), D(∇)Z•(l))Ω• .

(3.23)

Then term (3.22) is independent of Z(k) while a is an energy type functional. The energyfor the field Z(k) is just a(Z(k), Z(k)), and we can evaluate the off-diagonal elements of P ,i.e. Pkl for k 6= l, by the formula

2a(Z(k),Z(l)) = a(Z(k), Z(k)) + a(Z(l),Z(l))− a(Z(k) −Z(l), Z(k) −Z(l)) . (3.24)

It means, that all the entries of the matrix P are can be calculated using Lemma 3.2 and therules for derivatives of the energy functional. To this end we only need strong convergence ofthe minimizers as t → 0 which was shown in Corollary 3.4. If the data of the problem (1.6) -(1.8) are given through restrictions of sufficiently smooth data to the variable domains, thenthe methods of [57] for the first order shape sensitivity analysis can be applied. In the sameway it is possible to prove the existence of strong material derivatives for the minimizers,which is here omitted, since for the the shape differentiability of the energy type functionalsthe strong convergence of unique minimizers is sufficient.

To determine the shape derivatives of the entries of polarization matrix, we first considerthe entries on the diagonal, to this end we fix the index (k) and put

Z := Z(k),

then Z is the minimizer of the functional (3.1) for fixed interface Γ, where f and g1 aredefined in (2.25). Let Γt be defined as in (3.2). Using the representation (3.21) and (3.22)with k = l, it comes down to find the shape derivative of

J(Γ) := 2a(Z , Z ) := 2a(Z(k), Z(k)),

which we rewrite as follows (cf (2.25) and (3.1))

J(Γ) = a(Z ,Z )− 2L(Z ) = infv∈V 1

0

[a(v, v)− 2L(v)].

We repeat the same notation in the variable domain setting, hence

J(Γt) := 2at(Zt, Zt) = at(Zt, Zt)− 2Lt(Zt) = infvt

[at(vt, vt)− 2Lt(vt)].

22

To transport the minimized functional to the fixed domain, we replace simply the testfunctions vt by test functions of a special form useful for our purpose, namely vt := u T−1

t .In this way we get very simple expression to be differentiated

J(Γt) := infu

[at(u, u)− 2Lt(u)]

where at(u, u) and Lt(u) are the forms transported to the fixed domain and interface, re-spectively. It is not difficult to find the derivatives with respect to t of the both forms, whichwe denote a and L, respectively. Thus, the energy functional is shape differentiable and weget the expression

dJ(Γ;V ) = a(Z ,Z )− 2L(Z )

for its shape derivative. To be more precise, we get the shape derivatives of Pkk for allk. The expression of this particular form is already very useful for the numerical analysis,however, using the structure theorem we can also identify the boundary expression for theshape derivative, assuming the necessary regularity of the minimizers and of the interface.

The second step is just to extend the analysis of the shape differentiability for the ar-bitrary indices (k), (l) which can be performed in the same way in view of the formula(3.24). It means that we get exactly the same form of shape derivatives of all entries of thepolarization matrix as for the entries on the diagonal with (k) = (l).

3.5 Energy matrix function

We denote by J(Γt) the 6 × 6-matrix, where the entries Jk,l(Γt) are defined with the helpof Zt,(k) in the perturbed domains, i.e.

Jk,l(Γt) =12at(Zt,(k), Zt,(l))− Lt(Zt,(l)) = −1

2at(Zt,(k), Zt,(l)) =

=12(AD(∇)Zt,(k), D(∇)Zt,(l))Ωt +

12(A•D(∇)Z•t,(k), D(∇)Z•t,(l))Ω•t−

−(ft, Zt,(l))Ωt − (f•t , Z•t,(l))Ω•t − (g1t , Zt,(l) = Z•t,(l))Γt =

−12(AD(∇)Zt,(k), D(∇)Zt,(l))Ωt −

12(A•D(∇)Z•t,(k), D(∇)Z•t,(l))Ω•t

where ft = f(l)Tt, g1t = g1

(l)Tt and f(l), g1(l) are taken from (2.25). Assuming for simplicity

that the matrices A,A• are constant, we transport Jk,l(Γt) to the fixed domain which leadsto the following expression

J(Γt) =12at(Z t

(k),Zt(l))− Lt(Z t

(l)) = −12at(Z t

(k), Zt(l)) = (3.25)

=12(ϑ(t)ADt(∇)Zt

(k), Dt(∇)Zt

(l))Ω +12(ϑ(t)A•Dt(∇)Z•,t(k), D

t(∇)Z•,t(l) )Ω•−−(θ(t)F t, Zt

(l))Ω − (θ(t)F •t , Z•,t(l) )Ω• − (θ(t)g1,t, Zt(l))Γ =

−12(ϑ(t)ADt(∇)Zt

(k), Dt(∇)Zt

(l))Ω −12(ϑ(t)A•Dt(∇)Z•,t(k), D

t(∇)Z•,t(l) )Ω• ,

where again Dt(∇) = D(T−>∇x) according to the transformation rules. From Lemma 3.4it follows that in the fixed domain setting we have the shape differentiability result.

Proposition 3.5 All entries Jk,l(Γ) of the matrix shape functional J(Γt) are shape differ-entiable i.e., the following limits exist

dJk,l(Γ; V ) = limt→0

1t(Jk,l(Γt)− Jk,l(Γ)) .

Furthermore, the Hadamard structure theorem of the shape gradient [57] implies the conti-nuity of the mapping C1

0 (R3) 3 V 7→ dJk,l(Γ;V ) ∈ R. Therefore, there are shape gradientsgΓ

k,l such thatdJk,l(Γ; V ) = 〈gΓ

k,l, V · n〉Γ (3.26)

in the duality pairing on the interface Γ.

23

The direct evaluation of the shape derivative dJ(Γ;V ) leads to quite complicated expres-sions, however, explicit formulae are useful from the point of view of possible applications.We are going to present the formal time derivative at t = 0 of the energy functionals wherewe suppose regularity assumptions under which the surface integrals make sense,

dJk,l(Γ;V ) :=d

dtJk,l(Γt) =

12

∫

Γ

(AD(∇)Z(k), D(∇)Z(l))− (A•D(∇)Z•(k), D(∇)Z•(l))V · ndsx

−∫

Γ

(f − f•, Z(l))V · ndsx −∫

Γ

(θ′(0)g1 + (g1)′Γ), Z(l))V · ndsx ,

where (g1)′Γ is the so-called boundary shape derivative [57] of the element g1.If there is no regularity required here, the shape derivative dJ(Γ;V ) can be expressed

in terms of material derivatives by volume integrals, and the structure theorem [57] can beused in order to obtain the boundary formula in the sense of distributions.

3.6 Polarization tensor

The shape differentiability of energy functionals implies the shape differentiability of thepolarization tensor. From formula (2.30) we get for the perturbed interface

Pt = −∫

Ω•t

(A0 −A•(x)

)dx +

∫

Ωt

Ae(x) + 2J(Γt) . (3.27)

Therefore, the time derivative P ′ of the tensor Pt is given by the formula

P ′ = −∫

Γ

(A0 −A•(x) + Ae(x)

)V · ndsx + 2dJ(Γ; V ) . (3.28)

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